The question goes as follows:
Let $f : [a, b] → \mathbb{R}$ be a bounded function. Suppose thatthere is a partition $P$ of $[a, b]$ such that $L(P, f ) = U(P, f)$.Show that f is a constant function.
I understood the approach to show $m_i = M_i$ for the partition given considering intervals $[x_{i-1}, x_i]$.
We can do the same for the Greatest Integer Function as well, considering partitions of unit length starting from 0, but we can't say that it's constant too, since for certain other intervals that$M_i \neq M_{i+1}$, such as $[\frac{n}{2} , \frac{n+1}{2}]$.
How do we prove that $M_i = M_{i+1}$ as well, besides proving $m_i=M_i$,or if possible, bypass the above mentioned and still prove the function constant?
Clarification: I wish to know if it is necessary, and possible, to show that every subsequent partition has the same value as its infimum/supremum.